Diagrams.Backend.Cairo.Internal:setTexture from diagrams-cairo-1.3.0.3

Average Error: 12.9 → 0.4

Time: 7.9s

Precision: binary64

Cost: 2640

Math

\[\frac{x \cdot \left(y - z\right)}{y} \]

↓

\[\begin{array}{l} t_0 := \frac{x \cdot \left(y - z\right)}{y}\\ t_1 := x - \frac{x}{\frac{y}{z}}\\ \mathbf{if}\;t_0 \leq -\infty:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t_0 \leq -2 \cdot 10^{+85}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;t_0 \leq 5 \cdot 10^{+14}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t_0 \leq 2 \cdot 10^{+293}:\\ \;\;\;\;x - \frac{1}{\frac{y}{x \cdot z}}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(1 - \frac{z}{y}\right)\\ \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 (/ (* x (- y z)) y))

↓

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (/ (* x (- y z)) y)) (t_1 (- x (/ x (/ y z)))))
   (if (<= t_0 (- INFINITY))
     t_1
     (if (<= t_0 -2e+85)
       t_0
       (if (<= t_0 5e+14)
         t_1
         (if (<= t_0 2e+293)
           (- x (/ 1.0 (/ y (* x z))))
           (* x (- 1.0 (/ z y)))))))))

C

double code(double x, double y, double z) {
	return (x * (y - z)) / y;
}

↓

double code(double x, double y, double z) {
	double t_0 = (x * (y - z)) / y;
	double t_1 = x - (x / (y / z));
	double tmp;
	if (t_0 <= -((double) INFINITY)) {
		tmp = t_1;
	} else if (t_0 <= -2e+85) {
		tmp = t_0;
	} else if (t_0 <= 5e+14) {
		tmp = t_1;
	} else if (t_0 <= 2e+293) {
		tmp = x - (1.0 / (y / (x * z)));
	} else {
		tmp = x * (1.0 - (z / y));
	}
	return tmp;
}

Java

public static double code(double x, double y, double z) {
	return (x * (y - z)) / y;
}

↓

public static double code(double x, double y, double z) {
	double t_0 = (x * (y - z)) / y;
	double t_1 = x - (x / (y / z));
	double tmp;
	if (t_0 <= -Double.POSITIVE_INFINITY) {
		tmp = t_1;
	} else if (t_0 <= -2e+85) {
		tmp = t_0;
	} else if (t_0 <= 5e+14) {
		tmp = t_1;
	} else if (t_0 <= 2e+293) {
		tmp = x - (1.0 / (y / (x * z)));
	} else {
		tmp = x * (1.0 - (z / y));
	}
	return tmp;
}

Python

def code(x, y, z):
	return (x * (y - z)) / y

↓

def code(x, y, z):
	t_0 = (x * (y - z)) / y
	t_1 = x - (x / (y / z))
	tmp = 0
	if t_0 <= -math.inf:
		tmp = t_1
	elif t_0 <= -2e+85:
		tmp = t_0
	elif t_0 <= 5e+14:
		tmp = t_1
	elif t_0 <= 2e+293:
		tmp = x - (1.0 / (y / (x * z)))
	else:
		tmp = x * (1.0 - (z / y))
	return tmp

Julia

function code(x, y, z)
	return Float64(Float64(x * Float64(y - z)) / y)
end

↓

function code(x, y, z)
	t_0 = Float64(Float64(x * Float64(y - z)) / y)
	t_1 = Float64(x - Float64(x / Float64(y / z)))
	tmp = 0.0
	if (t_0 <= Float64(-Inf))
		tmp = t_1;
	elseif (t_0 <= -2e+85)
		tmp = t_0;
	elseif (t_0 <= 5e+14)
		tmp = t_1;
	elseif (t_0 <= 2e+293)
		tmp = Float64(x - Float64(1.0 / Float64(y / Float64(x * z))));
	else
		tmp = Float64(x * Float64(1.0 - Float64(z / y)));
	end
	return tmp
end

MATLAB

function tmp = code(x, y, z)
	tmp = (x * (y - z)) / y;
end

↓

function tmp_2 = code(x, y, z)
	t_0 = (x * (y - z)) / y;
	t_1 = x - (x / (y / z));
	tmp = 0.0;
	if (t_0 <= -Inf)
		tmp = t_1;
	elseif (t_0 <= -2e+85)
		tmp = t_0;
	elseif (t_0 <= 5e+14)
		tmp = t_1;
	elseif (t_0 <= 2e+293)
		tmp = x - (1.0 / (y / (x * z)));
	else
		tmp = x * (1.0 - (z / y));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := N[(N[(x * N[(y - z), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]

↓

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(x * N[(y - z), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]}, Block[{t$95$1 = N[(x - N[(x / N[(y / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, (-Infinity)], t$95$1, If[LessEqual[t$95$0, -2e+85], t$95$0, If[LessEqual[t$95$0, 5e+14], t$95$1, If[LessEqual[t$95$0, 2e+293], N[(x - N[(1.0 / N[(y / N[(x * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x * N[(1.0 - N[(z / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]

Target

Original	12.9
Target	3.3
Herbie	0.4

\[\begin{array}{l} \mathbf{if}\;z < -2.060202331921739 \cdot 10^{+104}:\\ \;\;\;\;x - \frac{z \cdot x}{y}\\ \mathbf{elif}\;z < 1.6939766013828526 \cdot 10^{+213}:\\ \;\;\;\;\frac{x}{\frac{y}{y - z}}\\ \mathbf{else}:\\ \;\;\;\;\left(y - z\right) \cdot \frac{x}{y}\\ \end{array} \]

Derivation

1. Split input into 4 regimes

2. if (/.f64 (*.f64 x (-.f64 y z)) y) < -inf.0 or -2e85 < (/.f64 (*.f64 x (-.f64 y z)) y) < 5e14

   1. Initial program 13.3

      \[\frac{x \cdot \left(y - z\right)}{y} \]

   2. Simplified 2.5

      \[\leadsto \color{blue}{x - z \cdot \frac{x}{y}} \]
      Proof
      (-.f64 x (*.f64 z (/.f64 x y))): 0 points increase in error, 0 points decrease in error
      (-.f64 (Rewrite<= *-lft-identity_binary64 (*.f64 1 x)) (*.f64 z (/.f64 x y))): 0 points increase in error, 0 points decrease in error
      (-.f64 (*.f64 (Rewrite<= *-inverses_binary64 (/.f64 y y)) x) (*.f64 z (/.f64 x y))): 0 points increase in error, 0 points decrease in error
      (-.f64 (Rewrite=> associate-*l/_binary64 (/.f64 (*.f64 y x) y)) (*.f64 z (/.f64 x y))): 67 points increase in error, 0 points decrease in error
      (-.f64 (Rewrite<= associate-*r/_binary64 (*.f64 y (/.f64 x y))) (*.f64 z (/.f64 x y))): 43 points increase in error, 63 points decrease in error
      (Rewrite=> distribute-rgt-out--_binary64 (*.f64 (/.f64 x y) (-.f64 y z))): 0 points increase in error, 3 points decrease in error
      (Rewrite=> associate-*l/_binary64 (/.f64 (*.f64 x (-.f64 y z)) y)): 87 points increase in error, 69 points decrease in error

   3. Taylor expanded in z around 0 5.7

      \[\leadsto x - \color{blue}{\frac{z \cdot x}{y}} \]

   4. Simplified 0.3

      \[\leadsto x - \color{blue}{x \cdot \frac{z}{y}} \]
      Proof
      (*.f64 x (/.f64 z y)): 0 points increase in error, 0 points decrease in error
      (Rewrite<= *-commutative_binary64 (*.f64 (/.f64 z y) x)): 0 points increase in error, 0 points decrease in error
      (Rewrite=> associate-*l/_binary64 (/.f64 (*.f64 z x) y)): 51 points increase in error, 58 points decrease in error

   5. Applied egg-rr 0.3

      \[\leadsto x - \color{blue}{\frac{x}{\frac{y}{z}}} \]

   if -inf.0 < (/.f64 (*.f64 x (-.f64 y z)) y) < -2e85

   1. Initial program 0.2

      \[\frac{x \cdot \left(y - z\right)}{y} \]

   if 5e14 < (/.f64 (*.f64 x (-.f64 y z)) y) < 1.9999999999999998e293

   1. Initial program 0.2

      \[\frac{x \cdot \left(y - z\right)}{y} \]

   2. Simplified 7.0

      \[\leadsto \color{blue}{x - z \cdot \frac{x}{y}} \]
      Proof
      (-.f64 x (*.f64 z (/.f64 x y))): 0 points increase in error, 0 points decrease in error
      (-.f64 (Rewrite<= *-lft-identity_binary64 (*.f64 1 x)) (*.f64 z (/.f64 x y))): 0 points increase in error, 0 points decrease in error
      (-.f64 (*.f64 (Rewrite<= *-inverses_binary64 (/.f64 y y)) x) (*.f64 z (/.f64 x y))): 0 points increase in error, 0 points decrease in error
      (-.f64 (Rewrite=> associate-*l/_binary64 (/.f64 (*.f64 y x) y)) (*.f64 z (/.f64 x y))): 67 points increase in error, 0 points decrease in error
      (-.f64 (Rewrite<= associate-*r/_binary64 (*.f64 y (/.f64 x y))) (*.f64 z (/.f64 x y))): 43 points increase in error, 63 points decrease in error
      (Rewrite=> distribute-rgt-out--_binary64 (*.f64 (/.f64 x y) (-.f64 y z))): 0 points increase in error, 3 points decrease in error
      (Rewrite=> associate-*l/_binary64 (/.f64 (*.f64 x (-.f64 y z)) y)): 87 points increase in error, 69 points decrease in error

   3. Applied egg-rr 0.2

      \[\leadsto x - \color{blue}{\frac{1}{\frac{y}{z \cdot x}}} \]

   if 1.9999999999999998e293 < (/.f64 (*.f64 x (-.f64 y z)) y)

   1. Initial program 59.2

      \[\frac{x \cdot \left(y - z\right)}{y} \]

   2. Simplified 1.8

      \[\leadsto \color{blue}{x - z \cdot \frac{x}{y}} \]
      Proof
      (-.f64 x (*.f64 z (/.f64 x y))): 0 points increase in error, 0 points decrease in error
      (-.f64 (Rewrite<= *-lft-identity_binary64 (*.f64 1 x)) (*.f64 z (/.f64 x y))): 0 points increase in error, 0 points decrease in error
      (-.f64 (*.f64 (Rewrite<= *-inverses_binary64 (/.f64 y y)) x) (*.f64 z (/.f64 x y))): 0 points increase in error, 0 points decrease in error
      (-.f64 (Rewrite=> associate-*l/_binary64 (/.f64 (*.f64 y x) y)) (*.f64 z (/.f64 x y))): 67 points increase in error, 0 points decrease in error
      (-.f64 (Rewrite<= associate-*r/_binary64 (*.f64 y (/.f64 x y))) (*.f64 z (/.f64 x y))): 43 points increase in error, 63 points decrease in error
      (Rewrite=> distribute-rgt-out--_binary64 (*.f64 (/.f64 x y) (-.f64 y z))): 0 points increase in error, 3 points decrease in error
      (Rewrite=> associate-*l/_binary64 (/.f64 (*.f64 x (-.f64 y z)) y)): 87 points increase in error, 69 points decrease in error

   3. Taylor expanded in x around 0 1.5

      \[\leadsto \color{blue}{\left(1 - \frac{z}{y}\right) \cdot x} \]
3. Recombined 4 regimes into one program.

4. Final simplification 0.4

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x \cdot \left(y - z\right)}{y} \leq -\infty:\\ \;\;\;\;x - \frac{x}{\frac{y}{z}}\\ \mathbf{elif}\;\frac{x \cdot \left(y - z\right)}{y} \leq -2 \cdot 10^{+85}:\\ \;\;\;\;\frac{x \cdot \left(y - z\right)}{y}\\ \mathbf{elif}\;\frac{x \cdot \left(y - z\right)}{y} \leq 5 \cdot 10^{+14}:\\ \;\;\;\;x - \frac{x}{\frac{y}{z}}\\ \mathbf{elif}\;\frac{x \cdot \left(y - z\right)}{y} \leq 2 \cdot 10^{+293}:\\ \;\;\;\;x - \frac{1}{\frac{y}{x \cdot z}}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(1 - \frac{z}{y}\right)\\ \end{array} \]

Alternatives

Alternative 1
Error	0.4
Cost	2512

\[\begin{array}{l} t_0 := \frac{x \cdot \left(y - z\right)}{y}\\ t_1 := x - \frac{x}{\frac{y}{z}}\\ \mathbf{if}\;t_0 \leq -\infty:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t_0 \leq -2 \cdot 10^{+85}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;t_0 \leq 9 \cdot 10^{-59}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t_0 \leq 2 \cdot 10^{+293}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(1 - \frac{z}{y}\right)\\ \end{array} \]

Alternative 2
Error	19.6
Cost	912

\[\begin{array}{l} t_0 := \frac{-z}{\frac{y}{x}}\\ \mathbf{if}\;y \leq -4.2823194274122806 \cdot 10^{+30}:\\ \;\;\;\;x\\ \mathbf{elif}\;y \leq 1.1 \cdot 10^{-170}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq 10^{-97}:\\ \;\;\;\;x\\ \mathbf{elif}\;y \leq 3.1 \cdot 10^{-43}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 3
Error	3.7
Cost	580

\[\begin{array}{l} \mathbf{if}\;z \leq -1 \cdot 10^{+150}:\\ \;\;\;\;\frac{-z}{\frac{y}{x}}\\ \mathbf{else}:\\ \;\;\;\;x - \frac{x}{\frac{y}{z}}\\ \end{array} \]

Alternative 4
Error	2.2
Cost	580

\[\begin{array}{l} \mathbf{if}\;z \leq -1 \cdot 10^{+33}:\\ \;\;\;\;x - \frac{z}{\frac{y}{x}}\\ \mathbf{else}:\\ \;\;\;\;x - \frac{x}{\frac{y}{z}}\\ \end{array} \]

Alternative 5
Error	25.2
Cost	64

\[x \]

Reproduce

herbie shell --seed 2022316 
(FPCore (x y z)
  :name "Diagrams.Backend.Cairo.Internal:setTexture from diagrams-cairo-1.3.0.3"
  :precision binary64

  :herbie-target
  (if (< z -2.060202331921739e+104) (- x (/ (* z x) y)) (if (< z 1.6939766013828526e+213) (/ x (/ y (- y z))) (* (- y z) (/ x y))))

  (/ (* x (- y z)) y))